Our result also opens up an alternative approach to establish the
GPT framework as a whole. In Section 
we reviewed the derivation of the framework from operational principles
which motivates the structure of state spaces in GPTs.
One may, however, arrive at the same framework by operationally motivating
the structure of effect spaces in GPTs (as convex, compact
subsets of a real vector space containing the zero vector, and a vector
\(u\)  such that \(u-e\)  is in the set for every effect \(e\) ),
combined with a minimal set of observables formed from the effects
in the space (namely, two-outcome observables and their convex combinations,
see Section  and [1]}, [2]}).
At this point a corollary to our Gleason-type theorem may be used
to indeed recover the structure of the state space (as a subset of
\(W\left(\mathcal {E}\right)\)  where \(\mathcal {E}\)  is the effect space
of the model) by simply assuming that a state must have a corresponding
frame function on this minimal set of observables.
